Proof of Independence of U from V & P with Respect to T

[mahdert]

Homework Statement

Show that the internal energy of a material whose equation of state has the form p = f(V), T is independent of the volume and the pressure. That is

\left(\frac{\partial U}{\partial V}\right)_{T} = 0

\left(\frac{\partial U}{\partial p}\right)_{T} = 0

Homework Equations

TdS = dU + pdV

The Attempt at a Solution

I know the answer intiutively, i just don't know how one would go about showing it.
I assume that U = f(p,v) and then take the partial derivatives, but I do not see where T comes into play

 

[RTW69]

Hmmm, Divide through by dv to get T ds/dv= dU/dv + p I am guessing you need to use a maxwell relation to get ds/dv to dp/dT I don't see how you get the partials to equal 0 though.

 

[mahdert]

I am trying to use the fundamental relation: Tds = dU + pdV and equate it with the partial expansion of U = U(V,p), however, I keep getting stuck because I do not know how to get the relationship of the change in U with either a change in V or p for a given temperature.

 

[RTW69]

not sure this helps but If U=U(V,p) you can use the chain rule to get to dU= (partial U/ partial V) dU + (partial U/partial p) dp

 

[mahdert]

Simple Thermodynamics Problem

Is this a single variable problem?